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where
D
(
x
) denotes the diffusion coefficients which depend on the structure of
space,
/
t
the differential operator with respect to the time variable and
∂
∂
@
@
@
@
@
@
r¼
x
1
;
x
2
; ...;
(3.12)
x
n
the differential operator with respect to the space variables. Here we use the inner
product
a
b
¼
a
1
b
1
þ
a
2
b
2
þþ
a
n
b
n
(3.13)
n
. The
diffusion equation (
3.11
) is a partial differential equation (PDE) of time and space
with nonlinear forms.
If there is a positive constant
D
0
such that
D
(
x
)
n
and
b
with two vectors, say
a
¼
(
a
1
,
a
2
,
...
,
a
n
)
¼
(
b
1
,
b
2
,
...
,
b
n
)
∈R
∈R
n
, then we
¼
D
0
for any
x
∈ R
have the linear form
@
@
t
u
ð
t
;
x
Þ¼
D
0
D
u
ð
t
;
x
Þ;
(3.14)
n
.
of a diffusion equation, where
D ¼r
·
r
is the Laplace operator in
R
Fourier analysis tells us the solution to (
3.14
) with initial data
u
(0
,x
)
¼
u
0
(
x
) for
n
,
any
x
∈ R
2
Z
e
ixx
e
t
jxj
2
n
pÞ
u
ð
t
;
x
Þ¼ð
2
u
0
ðxÞ
^
d
x;
(3.15)
4
D
0
n
R
where
^
u
0
stands for the Fourier transform of
u
0
with respect to space variables
2
Z
n
pÞ
e
ixx
u
0
ð
u
0
ðxÞ¼ð
^
2
x
Þ
dx
:
(3.16)
n
R
It is also possible to define calculus on networks. See Chap. 2 in Cardanobile
(
2010
) for integration by parts on networks.
3.5.3 Diffusion on Networks
In this section, we consider the diffusion phenomena of information on the variety
of networks, such as complete, random, stochastic and scale-free networks. We
provide each definition of the corresponding models of networks. The dynamics of
diffusion or percolation depends on the structure of networks. We see the property
of networks under the definition, and consider the characteristics of each network.
These results are a natural generalization of the previous works by Dan (
2011a
).
